prove in any triangle abc if one angle is 120 degree the triangle formed by feet of angle bisectors is right angled
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prove in any triangle abc if one angle is 120 degree the triangle formed by feet of angle bisectors is right angled
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Step-by-step explanation:
Let Triangle ABC, ∠A=120∘
Let AD, BE, CF are angle bisectors of A, B, C respectively.
In ΔABD, AC is exterior angles bisector (as ∠DAC=∠CAX=60∘).
BE is internal angle bisector meet at E . E is ex-center of ΔABD
So DE is exterior angle bisector of ΔABD
Then, ∠ADE=∠EDC,∠ADC=2∠ADE.
Similarly we can prove that for the ΔABC, F is ex-center as AB is exterior angle bisector and CF is interior angle bisector.
Hence DF is exterior angle bisector.
∠ADF=∠FDB,∠BDA=2∠ADF
BC is straight angle.
∠BDA+∠ADC=180∘
2∠ADF+2∠ADE=180∘
∠ADF+∠ADE=90∘
Hence DEF is right triangle.
Therefore the triangle formed by the feet’s of angle bisectors of triangle whose one angle measure is 120∘ is Right Triangle.
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